Animated S-Stacks in Visualization

• Animated S-Stacks are sequenced animated transitions connecting scatter plots from multivariate datasets while preserving point identity.
• They employ techniques such as spline interpolation and rotation transformations to maintain smooth perceptual continuity.
• Design guidance emphasizes point traceability and optimal transition timing in advanced multivariate visualization.

Searching arXiv for the cited papers and topic terminology to ground the article. Animated S-Stacks denote sequenced, stacked animated transitions between scatter plots drawn from different $2$D projections of the same multivariate dataset. In this usage, each stack element is a scatter plot panel, such as a SPLOM cell, a grand tour frame, or a DR-derived projection, and the stack is the ordered set of panels traversed by an analyst. The central technical problem is correspondence preservation: animation is used to maintain a mental map by moving points smoothly between successive panels during axis swaps or projection changes (Rodrigues et al., 2024). In a broader visualization-systems sense, the same phrase can also refer to staged, stacked-in-time transition specifications assembled from animation steps or keyframes, as formalized by Gemini and Gemini2 for single-view statistical graphics (Kim et al., 2020, Kim et al., 2021). A distinct mathematical usage appears in higher and derived geometry, where “animated $S$-stacks” means stacks of anima on a site $S$; that meaning belongs to homotopy theory and logarithmic geometry rather than information visualization (Hörmann, 2021, Zhang, 21 Jan 2026). Context therefore determines whether the term refers to animated multiview scatter-plot navigation, staged chart transitions, or stacks valued in spaces.

1. Definition and terminological scope

In multivariate visualization, Animated S-Stacks are sequenced, stacked animated transitions between scatter plots of different dimension pairs from the same underlying dataset. A transition maps one panel’s axes $D_1 \times D_2$ to the next panel’s axes $D_3 \times D_4$, preserving point identity across the sequence (Rodrigues et al., 2024). This formulation naturally covers SPLOM navigation, grand-tour frame-to-frame changes, and transitions to or from DR-derived projections.

The paper "Comparative Evaluation of Animated Scatter Plot Transitions" identifies six transition techniques that can serve as the per-step animation in such a stack: spline-based straight lines (STR), bundled splines (BUN), time-offset splines (TIM), and rotation-based staged rotation (STA), perspective rotation (PER), and orthographic rotation (ORT) (Rodrigues et al., 2024). Within this framework, a stack is not a static arrangement but an ordered traversal of views, with each step designed to preserve correspondence and support traceability.

A second, related meaning arises in declarative animation systems. Gemini defines transitions as compositions of steps over visual components such as marks, axes, legends, view, and pause, with synchronization and concatenation operators that permit staged animated sequences (Kim et al., 2020). Gemini2 extends this to keyframe-oriented transition design, treating an animation as a sequence of chart keyframes connected by per-pair animation specifications (Kim et al., 2021). This suggests a useful distinction between Animated S-Stacks as scatter-plot panel sequences and Animated S-Stacks as staged animation plans for statistical graphics more generally.

By contrast, in higher category theory and derived geometry, “animated $S$-stacks” means stacks of anima or spaces on a classical site $S$, satisfying descent conditions and modeled via simplicial presheaves or diagrammatic localizers (Hörmann, 2021). In logarithmic derived geometry, the phrase expands further to animated and spectral log stacks built from animated log rings and $\mathbb{E}_\infty$-log rings (Zhang, 21 Jan 2026). These are unrelated to visualization except by lexical overlap.

2. Transition models for scatter-plot stacks

The visualization literature underlying Animated S-Stacks distinguishes spline-based and rotation-based transitions. Spline-based methods directly interpolate point positions in $2$D screen space. STR uses linear interpolation from start $s=(x_0,y_0)$ to end $S$0 with base duration $S$1, constant speed, and an orthographic camera (Rodrigues et al., 2024). BUN uses cluster-aware cubic splines, with clusters at start and end detected via DBSCAN; points sharing a $S$2 pair follow bundled trajectories determined by common control points along the segment connecting cluster centroids $S$3 (Rodrigues et al., 2024). TIM retains clustered spline paths but executes cluster-pair groups sequentially in staggered temporal blocks, with total duration $S$4 of base (Rodrigues et al., 2024).

Rotation-based methods embed the transition in a $S$5D cube. For a $S$6D transition, the “new” axis is encoded as depth $S$7, then rotated about the unchanged axis so that depth is swapped into $S$8 or $S$9 (Rodrigues et al., 2024). A $S$0D transition is composed from two such $S$1D swaps in either horizontal-first (hf) or vertical-first (vf) order. STA consists of three sequential stages: orthographic-to-perspective camera morph to reveal depth, $S$2D cube rotation to swap axes, and perspective-to-orthographic camera morph to hide depth again, with total duration $S$3 of base (Rodrigues et al., 2024). PER overlaps perspective change and rotation continuously during a base-duration transition, whereas ORT performs pure $S$4D rotation under an orthographic camera so that $S$5 is never visually revealed (Rodrigues et al., 2024).

These techniques instantiate distinct perceptual hypotheses. Spline methods privilege explicit path continuity in the image plane, whereas rotation methods privilege a geometric model of axis exchange via a common $S$6D embedding. The empirical comparison in (Rodrigues et al., 2024) specifically evaluates their suitability for tracing individual points and clusters under ecologically realistic dot densities and overplotting.

3. Mathematical formulation and parameterization

The STR transition uses linear interpolation,

$S$7

which also describes stage morphs in staged rotation (Rodrigues et al., 2024). BUN and TIM use cubic Bézier or spline interpolation,

$S$8

where $S$9, $D_1 \times D_2$0, and $D_1 \times D_2$1 are cluster-defined control points (Rodrigues et al., 2024).

Rotation-based techniques rely on standard rotation matrices. The conceptual $D_1 \times D_2$2D rotation matrix is

$D_1 \times D_2$3

while the $D_1 \times D_2$4D cube embedding employs $D_1 \times D_2$5, $D_1 \times D_2$6, and $D_1 \times D_2$7 to realize axis swaps (Rodrigues et al., 2024). For $D_1 \times D_2$8D swaps, the rotation axis is the unchanged coordinate; for $D_1 \times D_2$9D swaps, two rotations are composed in either hf or vf order. The implementation in (Rodrigues et al., 2024) uses matrices, though quaternion-based spherical linear interpolation,

$D_3 \times D_4$0

is identified there as a robust alternative for smooth orientation blending.

Projection is either orthographic,

$D_3 \times D_4$1

with $D_3 \times D_4$2 ignored visually, or perspective,

$D_3 \times D_4$3

with focal length $D_3 \times D_4$4 or an equivalent perspective parameter blended over time (Rodrigues et al., 2024). STA introduces a staged depth-axis expansion schedule over total duration $D_3 \times D_4$5, with expansion, rotation, and collapse phases and a piecewise blend parameter $D_3 \times D_4$6 that mixes orthographic and perspective projections (Rodrigues et al., 2024). PER uses continuous co-evolution of a perspective blend parameter $D_3 \times D_4$7 and a rotation angle $D_3 \times D_4$8 over the same interval.

The paper fixes base duration at $D_3 \times D_4$9 for most methods, TIM at $S$0 of base, and STA at $S$1 of base, with constant speed and no variation in temporal distortion (Rodrigues et al., 2024). This fixed parameterization is important because the reported rankings concern a particular speed regime rather than an unconstrained family of animations.

4. Empirical evaluation of traceability

The evaluation in (Rodrigues et al., 2024) is a preregistered, within-subjects, crowdsourced user study with $S$2 participants, approximately $S$3 minutes per participant, attention checks, and exclusion criteria; rejected submissions were replaced to keep $S$4. The point traceability task required following one highlighted point across a single transition and clicking its final resting position; error was Euclidean distance in normalized plot coordinates $S$5 (Rodrigues et al., 2024). The cluster traceability task required following a highlighted cluster and classifying the interaction as remained, merged, or split, with proportion correct as the accuracy metric (Rodrigues et al., 2024).

The point task used the auto-mpg dataset with $S$6 points and $S$7 attributes under realistic overplotting conditions. The cluster task used a synthetic generative model with $S$8 points in $S$9 clusters of $S$0 points each plus $S$1 distractors, with controlled split, merge, and remain interactions (Rodrigues et al., 2024). The six animation techniques were tested across $S$2D and $S$3D transitions; for rotation-based $S$4D transitions, hf and vf orders were considered, whereas spline-based $S$5D transitions changed both axes simultaneously (bo) (Rodrigues et al., 2024). Each animation had $S$6 point tasks and $S$7 cluster tasks, with $S$8 training tasks with feedback prior to each block (Rodrigues et al., 2024).

Because Shapiro–Wilk indicated non-normality, the analysis used Wilcoxon signed-rank tests with Bonferroni correction and rank-biserial correlation $S$9 as effect size; subjective Likert ratings on speed, path clarity, and preference were analyzed with $\mathbb{E}_\infty$0 tests of independence (Rodrigues et al., 2024). This methodological choice matters because the principal findings are based on rank and distributional comparisons rather than Gaussian assumptions.

The main result is that rotation-based animations significantly outperform spline-based animations for point traceability: median error for rotations is $\mathbb{E}_\infty$1 versus $\mathbb{E}_\infty$2 for splines, with Wilcoxon exact $\mathbb{E}_\infty$3 and $\mathbb{E}_\infty$4 (Rodrigues et al., 2024). ORT and STA significantly outperform all other techniques for tracing individual points; PER and STR form a middle tier; BUN and TIM perform worst (Rodrigues et al., 2024). For cluster traceability, no significant differences were found across animations, and the overall mean accuracy was $\mathbb{E}_\infty$5, with ceiling effects suggested as an explanation (Rodrigues et al., 2024).

The study also reports direction and rotation-order effects. In the main study, $\mathbb{E}_\infty$6D vertical motion yielded lower error than horizontal, but a follow-up study reversed the pattern; similarly, in $\mathbb{E}_\infty$7D transitions vf outperformed hf in the main study, while hf outperformed vf in the follow-up, and bo was consistently worst (Rodrigues et al., 2024). The paper attributes these reversals to probable data-distribution confounds, such as discrete versus continuous axes and unique value counts (Rodrigues et al., 2024). A conservative conclusion is therefore that simultaneous both-axis swaps are undesirable for point tracing, while the optimal sequential order may depend on the distributional properties of the specific dimensions.

5. Design guidance and implementation patterns

The reported results yield explicit guidelines for designing Animated S-Stacks in scatter-plot systems. ORT is recommended by default for point-focused transitions because it yielded the best traceability and is simpler to implement than staged perspective variants (Rodrigues et al., 2024). STA is recommended when revealing depth helps explain the axis exchange, with a duration of approximately $\mathbb{E}_\infty$8 the base duration (Rodrigues et al., 2024). For $\mathbb{E}_\infty$9D swaps, sequential axis exchanges are preferred over simultaneous changes, and the first axis to swap should be chosen with attention to data characteristics, especially the number of continuous or varied values, to reduce overplotting during rotation (Rodrigues et al., 2024).

Additional guidance concerns timing and auxiliary visual encodings. The paper recommends a base duration of $2$0, constant speed, and smooth ease-in-out for camera blends and rotations while keeping per-point speed approximately constant within each stage (Rodrigues et al., 2024). Pre-animation highlighting of the target point or cluster is helpful; temporary halos or brief trails are proposed as plausible aids, but persistent trails are discouraged in cluttered views because they increase overdraw (Rodrigues et al., 2024). BUN and TIM are discouraged when point traceability is critical in high-density plots, and STR is described as acceptable if rotation is infeasible, though with lower expected accuracy than ORT or STA (Rodrigues et al., 2024).

The same paper provides a D3.js plug-in and demo code encapsulating STR, BUN, TIM, STA, PER, and ORT, including camera handling and path generation, with DBSCAN-based clustering options for BUN and TIM (Rodrigues et al., 2024). A typical pattern is data join followed by a transition call configured with mode, duration, order, easing, and clustering parameters. For S-Stacks across SPLOM panels, neighboring panels are traversed by iterating panel $2$1 to panel $2$2 transitions (Rodrigues et al., 2024). This operationalizes the concept of a stack as a reusable interaction pattern rather than a one-off animated effect.

Gemini and Gemini2 generalize these ideas from scatter plots to statistical graphics at large. Gemini defines a transition grammar in which a step is the basic unit of change over components such as marks, axes, legends, view, and pause; steps are combined with synchronization and concatenation operators to create staged transition plans (Kim et al., 2020). The specification formalism includes step, change, timing, staggering, and enumerator constructs, together with a ranking cost function,

$2$3

where $2$4 aggregates per-change weights, $2$5 is a sigmoid perceptual-capacity function, and $2$6 applies bundling penalties or discounts (Kim et al., 2020). These constructs provide a language for authoring animated step stacks with perceptual constraints such as object constancy, unavailable scales, unavailable encodings, and overflow avoidance.

Gemini2 moves from step composition to keyframe-oriented staging. It represents an animation as

$2$7

where each $2$8 is a Vega-Lite chart keyframe and each $2$9 is a Gemini animation spec (Kim et al., 2021). Candidate stage orders are scored by heuristic semantic rules,

$s=(x_0,y_0)$0

and full multi-pair animations are ranked by summed Gemini complexity,

$s=(x_0,y_0)$1

This supports semantically coherent staged transitions that Gemini alone cannot express, especially when transient intermediate keyframes are needed (Kim et al., 2021). A plausible implication is that the scatter-plot notion of an Animated S-Stack and the grammar/keyframe notion can be unified as two levels of abstraction: per-step geometric correspondence at the panel level and global stage orchestration at the system level.

6. Related meanings in higher and derived geometry

Outside visualization, “animated $s=(x_0,y_0)$2-stacks” has an established meaning in homotopy theory: stacks of anima, equivalently spaces or $s=(x_0,y_0)$3-groupoids, on a classical site $s=(x_0,y_0)$4 (Hörmann, 2021). An animated presheaf is a presheaf of anima, modeled as a simplicial presheaf up to weak equivalence, and an $s=(x_0,y_0)$5-stack is a presheaf $s=(x_0,y_0)$6 satisfying descent (Hörmann, 2021). For a covering family with Čech nerve $s=(x_0,y_0)$7, the descent condition is

$s=(x_0,y_0)$8

while hyperdescent requires the same for every hypercover (Hörmann, 2021). The paper "Higher stacks as diagrams" shows that the homotopy theory of non-hypercomplete animated $s=(x_0,y_0)$9-stacks can be presented either as the Čech-localized model category of simplicial presheaves or diagrammatically via a smallest localizer $S$00 on the $S$01-category $S$02 of diagrams in $S$03 (Hörmann, 2021).

The core equivalence is implemented by a nerve functor

$S$04

and a Grothendieck construction

$S$05

with natural weak equivalences $S$06 and $S$07 (Hörmann, 2021). Consequently, the non-hypercomplete $S$08-topos of animated $S$09-stacks admits both a simplicial-presheaf and a purely combinatorial diagrammatic presentation. This meaning of the term is categorical and unrelated to animated visualization, except that both domains are concerned with structured transitions between local views.

A further specialization appears in derived logarithmic geometry. The paper "Derived logarithmic deformation theory and moduli stacks of derived logarithmic structures" develops animated log rings, $S$10-log rings, spectral log stacks, and derived log Deligne–Mumford stacks (Zhang, 21 Jan 2026). An animated prelog ring $S$11 is an animated log ring if the map

$S$12

is an equivalence (Zhang, 21 Jan 2026). The paper constructs moduli functors

$S$13

establishes étale descent, cotangent complexes, and Artin representability, and defines $S$14-root stacks as inverse limits over root constructions (Zhang, 21 Jan 2026). Here again, “animated” means derived or simplicial enrichment rather than time-varying graphics.

The coexistence of these meanings creates an avoidable source of ambiguity. In visualization, Animated S-Stacks are about preserving perceptual correspondence across projections; in higher and log geometry, animated $S$15-stacks are sheaf-theoretic objects valued in spaces (Rodrigues et al., 2024, Hörmann, 2021, Zhang, 21 Jan 2026). Technical writing benefits from making the intended domain explicit.

7. Limitations, confounds, and open directions

For scatter-plot Animated S-Stacks, several limitations remain open. Cluster traceability showed no significant differences across transition types, likely because the chosen speed regime created ceiling effects in which clusters behaved as perceptual entities regardless of animation type (Rodrigues et al., 2024). The study fixed speed profiles and did not vary temporal distortion, leaving open whether alternative easing schedules or stage timing could differentiate cluster performance (Rodrigues et al., 2024). Direction effects reversed under axis swaps in a follow-up study of $S$16, suggesting sensitivity to data distributions and discretization rather than stable perceptual asymmetries (Rodrigues et al., 2024). Desktop-only evaluation, modest panel sizes, and within-subject sessions of approximately $S$17 minutes also limit generalization to mobile or prolonged analytical use (Rodrigues et al., 2024).

Gemini’s and Gemini2’s limitations are of a different kind. Gemini targets transitions between two single-view statistical graphics with at most one $S$18-axis and one $S$19-axis, and does not support richer multi-view or layered scenarios without extension (Kim et al., 2020). Some desired designs are not expressible because the grammar does not assign separate timing to distinct low-level properties within a single step, requiring decomposition into multiple steps (Kim et al., 2020). Gemini2 expands expressiveness through independent keyframes, but remains bounded by GraphScape’s single-view Cartesian scope and hand-tuned heuristic rule weights (Kim et al., 2021). This suggests that a full theory of Animated S-Stacks for modern dashboards or linked views remains incomplete.

In the higher-categorical usage, a principal open problem is extension from Čech-local to hypercover-local diagrammatic localizers. The paper (Hörmann, 2021) presents the smallest localizer $S$20 corresponding to Čech descent, but notes that an axiomatic characterization of the analogous hypercover-localizer remains open. In derived logarithmic geometry, the agenda shifts to deformation-theoretic representability, root-stack constructions, and cotangent-complex comparison after $S$21-completion (Zhang, 21 Jan 2026). These are unrelated research programs, but they underscore that the phrase “Animated S-Stacks” now spans several advanced literatures with incompatible definitions.

Taken together, the current evidence supports a narrow but clear conclusion for visualization practice: when the task is point traceability across a sequence of scatter-plot projections, Animated S-Stacks should preferentially use orthographic or staged rotations, avoid simultaneous two-axis swaps, and treat spline bundling or temporal offsets with caution in dense views (Rodrigues et al., 2024). At the systems level, declarative grammars and keyframe recommenders provide a complementary language for composing such steps into longer staged sequences (Kim et al., 2020, Kim et al., 2021). In mathematics, however, the same phrase denotes sheaf-theoretic stacks of anima or their logarithmic derived analogues, and the term must be interpreted accordingly (Hörmann, 2021, Zhang, 21 Jan 2026).